cnanfadi fu'ivla

x1 (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data x2 (completely specified ordered multiset/list) using function x3 (defaults according to the notes; if it is an extended-real number, then it has a particular interpretation according to the Notes) with weights x4 (completely specified ordered multiset/list with same cardinality/length as x2; defaults according to Notes).

Potentially dimensionful. Make sure to convert x3 from an operator to a sumti; x3 is the 'f' in "f-mean" and must be a complex-valued, single-valued function which is defined and continuous on x2 and which is injective; it defaults to the pth-power function (zp) for some nonzero p (note that it need not be positive or an integer) and indeterminate/variable/input z, or log, or exp (as functions); culture or context can further constrain the default. If x2 is set to "z+ \infty ." (the exponent is positive infinity, given by "ma'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result (x1) is the weight-sum-scaled maximum of the products of the data (terms of x2) with their corresponding weights (terms of x4) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if x2 is set to "z- \infty ." (the exponent is negative infinity, given by "ni'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result (x1) is the weight-sum-scaled minimum of the products of the data (terms of x2) with their corresponding weights (terms of x4) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of x4 is the ordered set of n terms with each term equal identically to 1/n, where the cardinality of x2 is n. Let "f" denote the sumti in x3, "yi" denote the ith term in x2 for all i, n denote the cardinality of x2 (thus also x4), and wi denote the ith term in x4 for any i; then the result x1 is equal to: f(-1)(Sum(wi f(y in Set(1,...,n)) / Sum(wi, i in Set(1,...,n))). Note that if the weights are all 1 and x2 is set equal to not the pth-power function, but instead the pth-power function left-composed with the absolute value function (or the forward difference function), then the result is the p-norm on x2 scaled by n(-1/p) for integer n being the cardinality of x2. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow x2 to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily x3. If x3 is a single extended-real number p (not a function), then this word refers to the weighted power-mean and it is equivalent to letting x2 equal the pth-power function as before iff p is nonzero real, the max or min as before if p is infinite (according to its signum as before), and log if p=0 (thus making the overall mean refer to the geometric mean); this overloading is for convenience of usage and will not cause confusion because constant functions are very much so not injective.


In notes:

cnanlagau
x1 is the generalized arithmetic-geometric mean of the elements of the 2-element set x2 (set; cardinality must be 2) of order x3 (either single extended-real number xor an unordered pair/2-element set of extended-real numbers).